Flows on quaternionic-Kahler and very special real manifolds

BPS solutions of 5-dimensional supergravity correspond to certain gradient flows on the product MxN of a quaternionic-Kahler manifold M of negative scalar curvature and a very special real manifold N of dimension ngreater than or equal to0. Such gradient flows are generated by the ``energy function'' f=P-2, where P is a (bundle-valued) moment map associated to n+1 Killing vector fields on M. We calculate the Hessian of f at critical points and derive some properties of its spectrum for general quaternionic-Kahler manifolds. For the homogeneous quaternionic-Kahler manifolds we prove more specific results depending on the structure of the isotropy group. For example, we show that there always exists a Killing vector field vanishing at a point p?M such that the Hessian of f at p has split signature. This generalizes results obtained recently for the complex hyperbolic plane (universal hypermultiplet) in the context of 5-dimensional supergravity. For symmetric quaternionic-Kahler manifolds we show the existence of non-degenerate local extrema of f, for appropriate Killing vector fields. On the other hand, for the non-symmetric homogeneous quaternionic-Kahler manifolds we find degenerate local minima.